Grid systems designed to hold building blocks in a spaced relationship are well known. Most commonly, these systems are intended as an alternative to laying the blocks (typically glass, plastic or masonry) in mortar. The existing grid systems tend to use framing member components of thin dimensions and of relatively weak construction. Furthermore, existing grid systems have relatively weak connections between the horizontal and vertical framing members and, in some cases, no connections at all, and do not include the capability to accept strengthening elements, such as steel rods, inserted within the framing members to provide added strength. In addition, existing grid systems typically rely on the block elements themselves to act as gauges to determine the spacing between the vertical or horizontal framing members. This can result in a grid system which is not precisely aligned owing to variations in the sizes of the blocks or foreign objects that may be lodged between the blocks and the framing members. Moreover, should water pass through the outer portions of the joints, existing grid systems do not have a provision for directing the leakage water to the bottom of the wall and then exhausting the water to a desired location. Finally, existing grid systems are not built with the option for accepting rectilinear blocks which have flat, square edges and alternatively, common glass blocks which have recessed ridges around their perimeter edges. Thus, there is a need for a block mounting system that addresses one or more of the above-described problems associated with existing grid systems.